I wrote, in trying to compare my condition of consistency-completeness with
Steve Simpson's related condition:
>Thus, in addition to Steve's question, I am also asking: Are there
>sentences S of PA such that (i) PA does not prove 'S-->Con(T)', (ii) PA |-
>GC--> S, and (iii) if PA |- S, then PA is inconsistent.
Torkel Franzen said in reply:
But there isn't any counterfactual conditional in this! (iii) is simply
equivalent to "S is unprovable in PA". So S satisfying (i)-(iii) exists
iff GC is not provable in PA, assuming T to refer to some theory for
which Con(T) is not provable in PA. (Take Q to be undecidable in both
PA+not-GC and PA+not-Con(T), and let S be GCvQ.)
Sorry, Torkel, but I intended (iii) to be read counterfactually. As I have
been saying, I don't think you can eliminate the counterfactual element. I
was just trying to be as clear as I could about the other differences
between Steve's condition and mine. I think we can discuss that without
having to agree on the eliminability/ineliminability of the counterfactual.
That's what I was trying to do. My heart isn't in these attempts, however,
since I don't see any reason to think that you can capture the (important
part of) the force of the counterfactual by asserting the provability of
the associated material conditional in some formal system. That's my basic
point and concern.
Robert Black writes:
In reply to Michael Detlefsen I am tempted to write pages on the exact
nature of and relations between material conditionals, English indicative
conditionals and English subjunctive conditionals.
I wish he WOULD respond to my concern, but nothing in this note does so. In
fact, the remark he makes about Bill Tait's example being better than mine
indicates that he really doesn't understand the problem at all.
Black says:
Hence it is completely natural that people have tried to make
sense of what Detlefsen says by providing a narrower semantics for the
subjunctive conditional; it would have to be one where the subjunctive
conditional 'If it were that A it would be that C' doesn't just come out as
automatically true when 'A & not-C' is necessarily false. And the only way
I can see to do this is indeed to talk about the derivability of C from A
in some suitably restricted formal system. The obvious candidate here, as
Vladimir Kanovei pointed out, is PA itself, and in that case Bill Tate's
example answers Detlefsen's question. (It's a better example than
Detlefsen's own neg-Cons(PA), since Bill's S is true.)
I don't deny that it is NATURAL (or at least common) to try to substitute a
condition like 'A-->C is provable in T' for a counterfactual ' if A then
C'. What I do deny is that it is plausible or convincing to do so. Maybe I
thought that this was more evident to people on this list than it is. So,
I'll try to lay out my reasons for this more carefully, using the example
of chief concern to me. The (type of) subjunctive or counterfactual
conditional of concern is this:
(#) If S is provable in PA, then PA is inconsistent
My position is and has been that (#) is true and assertable when known
undecidable sentences (e.g. godel sentences, the Ramsey sentence, etc.) are
substituted for S. My (consistency-completeness) question was whether this
is so for all S undecidable in PA.
Within a very short time, there were a number of people who contacted me
with essentially Vladimir Kanovei's suggestion; namely, that I surely
couldn't be meaning to ask the (#) type question, since that question isn't
clear. I must instead be meaning to ask whether:
(PA-#): PA |- Prov_PA(S) --> negCon(PA), for S undecidable in PA.
Vladimir and the others then went on to point out what I was already well
aware of: namely, that (PA-#) is not satisfied by such S as 'neg G' and
'neg Con(PA)'.
My reply was that I had thought of this. Indeed, my having thought of it
was my reason for rejecting (PA-#) as a substitute for (#). This remains my
position. I thought it was probably evident to everyone why. But Robert
Black's remark that Bill Tait's counterexample to (PA-#) is better than
mine (and also suggested to me in off-fom postings by Harvey and John
Steel) suggests that this is not so. I'll thus try to lay out the reasoning
in more detail.
Consider the instance of (#) obtained by substituting neg Con(PA) for S.
That's the example that Vladimir, I (and also Harvey and John Steel in
private communication) pointed to. Let's call that instance (neg
Con(PA)-#). I make the following claim:
(I) (neg Con(PA)-#) is true and assertable.
I take this as very strong evidence that (PA-#) cannot be substituted for
(#). The reason, which I had previously assumed was clear, is this:
(II) In order to substitute (PA-#) for (#), you have to believe that an
instance of 'Prov_PA(S) --> negCon(PA)' is provable in PA just in case the
corresponding instance of (#) is assertable.
(III) The instance of 'Prov_PA(S) --> negCon(PA)' (viz. 'Prov_PA(neg
Con(PA)) --> negCon(PA)') that corresponds to (neg Con(PA)-#) is not
provable in PA.
Therefore
CLAIM: (PA-#) is not a proper substitute for (#).
This is my position. If you disagree with it, then you must either deny (I)
or (II). I think that denying (I), or saying that it's not clear enough
either to affirm or deny, is a truly desperate measure. If you deny (II),
then I can only think that it would be because you think 'Prov_PA(S) -->
negCon(PA)' is not a good rendition in PA of the conditional in (#) (i.e.
if S is provable in PA, then PA is inconsistent). Fine, but what then IS a
good rendition? I see none ... though, Steve's post on the subject and a
couple of my later posts explore some possibilities. In the end, none of
the things I could think of allowed me to eliminate the counterfactual
element. (This inability to find a suitable PA formalization of
consistency-completeness was, by the way, a big part of my reason for
wanting to discuss it on FOM.)
Now as to the superiority of Bill Tait's counterexample over mine (and
Vladimir's and the rest). Bill's counterexample is fine ... in that it is a
counterexample. But the point was not to find a 'true' case. The point was
simply to an unprovable case. That's what's needed to produce the conflict
between (I) and (III). It's the UNPROVABILITY in PA of 'Prov_PA(neg
Con(PA)) --> negCon(PA)' that matters. Also--and this is the crucial
point--one needs a counterexample whose consequent is 'neg Con(PA)' since
the consequent of the corresponding instance of (#) is 'PA is
inconsistent'. So the counterexample I (and Vladimir and Harvey and John)
used is EXACTLY the counterexample that is wanted. It is better than Bill's
in that it makes the collision of (I) and (III) apparent. Still, I having
nothing against Bill's. One can make do with it too. It's just not as
direct as the one I used.
If anyone disagrees with what I have been saying about
consistency-completeness, I feel that they must be denying either (I) or
(II). It seems that Joe Shipman and Torkel Franzen want to deny (I). Fine,
if you can live with that, go ahead. I can't. I'm not sure what Robert
Black wants to do ... I don't see that he understood the problem. He
assures us, though, that (II) is an entirely reasonable thing to assert. I
agree with him that it is 'the obvious candidate'. Obviously, for the
simple reasons given above, I do not agree that it is a reasonable thing to
assert.
Can I explain what is going on here? Not in detail. But I will say this. So
long as one uses the usual provability expressions in PA (i.e. ones that
satisfy the Derivability Conditions), then conditionals of the form
'Prov_PA(S) --> S' will have the Lob property. I think that's why you can't
formalize (#) in PA. The English conditional (or conditional in the
language of thought) that appears in (#) does NOT have the Lob property. No
reading of the literature on counterfactuals will controvert that fact. So
long as that is so, I don't see that (#) can be expressed by a formula of
PA. That being so, I don't think that consistency-completeness can be
formalized in PA. That does not suggest to me that the problem is not real,
genuine or clear. Rather, it suggests that some questions--including some
fom questions--to which we'd like answers are not questions that can be
formulated in PA.
Mic Detlefsen
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Michael Detlefsen
Department of Philosophy
University of Notre Dame
Notre Dame, Indiana 46556
U.S.A.
e-mail: Detlefsen.1 at nd.edu
FAX: 219-631-8609
Office phone: 219-631-7534
Home phone: 219-232-7273
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